//private double DXDstrike_;
public AmericanPayoffAtExpiry(double spot, double discount, double dividendDiscount, double variance, StrikedTypePayoff payoff) {
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
if (!(spot_ > 0.0))
throw new ApplicationException("positive spot_ value required");
forward_ = spot_ * dividendDiscount_ / discount_;
if (!(discount_ > 0.0))
throw new ApplicationException("positive discount required");
if (!(dividendDiscount_ > 0.0))
throw new ApplicationException("positive dividend discount_ required");
if (!(variance_ >= 0.0))
throw new ApplicationException("negative variance_ not allowed");
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
// binary cash-or-nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null) {
K_ = coo.cashPayoff();
//DKDstrike_ = 0.0;
}
// binary asset-or-nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null) {
K_ = forward_;
//DKDstrike_ = 0.0;
mu_ += 1.0;
}
log_H_S_ = Math.Log(strike_ / spot_);
double n_d1;
double n_d2;
double cum_d1_;
double cum_d2_;
if (variance_ >= Const.QL_EPSILON) {
D1_ = log_H_S_ / stdDev_ + mu_ * stdDev_;
D2_ = D1_ - 2.0 * mu_ * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1 = f.derivative(D1_);
n_d2 = f.derivative(D2_);
} else {
if (log_H_S_ > 0) {
cum_d1_ = 1.0;
cum_d2_ = 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1 = 0.0;
n_d2 = 0.0;
}
switch (type) {
// up-and-in cash-(at-hit)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
case Option.Type.Call:
if (strike_ > spot_) {
alpha_ = 1.0 - cum_d2_; // N(-d2)
DalphaDd1_ = -n_d2; // -n( d2)
beta_ = 1.0 - cum_d1_; // N(-d1)
DbetaDd2_ = -n_d1; // -n( d1)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
// down-and-in cash-(at-hit)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
case Option.Type.Put:
if (strike_ < spot_) {
alpha_ = cum_d2_; // N(d2)
DalphaDd1_ = n_d2; // n(d2)
beta_ = cum_d1_; // N(d1)
DbetaDd2_ = n_d1; // n(d1)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
default:
throw new ApplicationException("invalid option type");
}
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) || (type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_) {
Y_ = 1.0;
X_ = 1.0;
//DYDstrike_ = 0.0;
//DXDstrike_ = 0.0;
} else {
Y_ = 1.0;
X_ = Math.Pow((double)(strike_ / spot_), (double)(2.0 * mu_));
// DXDstrike_ = ......;
}
}
// critical commodity price
//public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
// double variance, double tolerance = 1e-6);
public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
double variance, double tolerance)
{
// Calculation of seed value, Si
double n= 2.0*Math.Log(dividendDiscount/riskFreeDiscount)/(variance);
double m=-2.0*Math.Log(riskFreeDiscount)/(variance);
double bT = Math.Log(dividendDiscount/riskFreeDiscount);
double qu, Su, h, Si;
switch (payoff.optionType()) {
case Option.Type.Call:
qu = (-(n-1.0) + Math.Sqrt(((n-1.0)*(n-1.0)) + 4.0*m))/2.0;
Su = payoff.strike() / (1.0 - 1.0/qu);
h = -(bT + 2.0*Math.Sqrt(variance)) * payoff.strike() /
(Su - payoff.strike());
Si = payoff.strike() + (Su - payoff.strike()) *
(1.0 - Math.Exp(h));
break;
case Option.Type.Put:
qu = (-(n-1.0) - Math.Sqrt(((n-1.0)*(n-1.0)) + 4.0*m))/2.0;
Su = payoff.strike() / (1.0 - 1.0/qu);
h = (bT - 2.0*Math.Sqrt(variance)) * payoff.strike() /
(payoff.strike() - Su);
Si = Su + (payoff.strike() - Su) * Math.Exp(h);
break;
default:
throw new ArgumentException("unknown option type");
}
// Newton Raphson algorithm for finding critical price Si
double Q, LHS, RHS, bi;
double forwardSi = Si * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSi/payoff.strike()) + 0.5*variance) /
Math.Sqrt(variance);
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double K = (riskFreeDiscount!=1.0 ? -2.0*Math.Log(riskFreeDiscount)/
(variance*(1.0-riskFreeDiscount)) : 0.0);
double temp = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
switch (payoff.optionType()) {
case Option.Type.Call:
Q = (-(n-1.0) + Math.Sqrt(((n-1.0)*(n-1.0)) + 4 * K)) / 2;
LHS = Si - payoff.strike();
RHS = temp + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q) +
(1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS)/payoff.strike() > tolerance) {
Si = (payoff.strike() + RHS - bi * Si) / (1 - bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi/payoff.strike())+0.5*variance)
/Math.Sqrt(variance);
LHS = Si - payoff.strike();
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
RHS = temp2 + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q)
+ (1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance))
/ Q;
}
break;
case Option.Type.Put:
Q = (-(n-1.0) - Math.Sqrt(((n-1.0)*(n-1.0)) + 4 * K)) / 2;
LHS = payoff.strike() - Si;
RHS = temp - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount * cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS)/payoff.strike() > tolerance) {
Si = (payoff.strike() - RHS + bi * Si) / (1 + bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi/payoff.strike())+0.5*variance)
/Math.Sqrt(variance);
LHS = payoff.strike() - Si;
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance))*riskFreeDiscount;
RHS = temp2 - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount * cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
}
break;
default:
throw new ArgumentException("unknown option type");
}
return Si;
}
public AmericanPayoffAtHit(double spot, double discount, double dividendDiscount, double variance, StrikedTypePayoff payoff) {
spot_ = spot;
discount_ = discount;
dividendDiscount_ = dividendDiscount;
variance_ = variance;
if (!(spot_ > 0.0))
throw new ApplicationException("positive spot value required");
if (!(discount_ > 0.0))
throw new ApplicationException("positive discount required");
if (!(dividendDiscount_ > 0.0))
throw new ApplicationException("positive dividend discount required");
if (!(variance_ >= 0.0))
throw new ApplicationException("negative variance not allowed");
stdDev_ = Math.Sqrt(variance_);
Option.Type type = payoff.optionType();
strike_ = payoff.strike();
log_H_S_ = Math.Log(strike_ / spot_);
double n_d1;
double n_d2;
double cum_d1_;
double cum_d2_;
if (variance_ >= Const.QL_Epsilon) {
if (discount_ == 0.0 && dividendDiscount_ == 0.0) {
mu_ = -0.5;
lambda_ = 0.5;
} else if (discount_ == 0.0) {
throw new ApplicationException("null discount not handled yet");
} else {
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
lambda_ = Math.Sqrt(mu_ * mu_ - 2.0 * Math.Log(discount_) / variance_);
}
D1_ = log_H_S_ / stdDev_ + lambda_ * stdDev_;
D2_ = D1_ - 2.0 * lambda_ * stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_ = f.value(D2_);
n_d1 = f.derivative(D1_);
n_d2 = f.derivative(D2_);
} else {
// not tested yet
mu_ = Math.Log(dividendDiscount_ / discount_) / variance_ - 0.5;
lambda_ = Math.Sqrt(mu_ * mu_ - 2.0 * Math.Log(discount_) / variance_);
if (log_H_S_ > 0) {
cum_d1_ = 1.0;
cum_d2_ = 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_ = 0.0;
}
n_d1 = 0.0;
n_d2 = 0.0;
}
switch (type) {
// up-and-in cash-(at-hit)-or-nothing option
// a.k.a. american call with cash-or-nothing payoff
case Option.Type.Call:
if (strike_ > spot_) {
alpha_ = 1.0 - cum_d1_; // N(-d1)
DalphaDd1_ = -n_d1; // -n( d1)
beta_ = 1.0 - cum_d2_; // N(-d2)
DbetaDd2_ = -n_d2; // -n( d2)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
// down-and-in cash-(at-hit)-or-nothing option
// a.k.a. american put with cash-or-nothing payoff
case Option.Type.Put:
if (strike_ < spot_) {
alpha_ = cum_d1_; // N(d1)
DalphaDd1_ = n_d1; // n(d1)
beta_ = cum_d2_; // N(d2)
DbetaDd2_ = n_d2; // n(d2)
} else {
alpha_ = 0.5;
DalphaDd1_ = 0.0;
beta_ = 0.5;
DbetaDd2_ = 0.0;
}
break;
default:
throw new ApplicationException("invalid option type");
}
muPlusLambda_ = mu_ + lambda_;
muMinusLambda_ = mu_ - lambda_;
inTheMoney_ = (type == Option.Type.Call && strike_ < spot_) || (type == Option.Type.Put && strike_ > spot_);
if (inTheMoney_) {
forward_ = 1.0;
X_ = 1.0;
DXDstrike_ = 0.0;
} else {
forward_ = Math.Pow(strike_ / spot_, muPlusLambda_);
X_ = Math.Pow(strike_ / spot_, muMinusLambda_);
// DXDstrike_ = ......;
}
// Binary Cash-Or-Nothing payoff?
CashOrNothingPayoff coo = payoff as CashOrNothingPayoff;
if (coo != null) {
K_ = coo.cashPayoff();
DKDstrike_ = 0.0;
}
// Binary Asset-Or-Nothing payoff?
AssetOrNothingPayoff aoo = payoff as AssetOrNothingPayoff;
if (aoo != null) {
if (inTheMoney_) {
K_ = spot_;
DKDstrike_ = 0.0;
} else {
K_ = aoo.strike();
DKDstrike_ = 1.0;
}
}
}
// public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount = 1.0) {
public BlackCalculator(StrikedTypePayoff payoff, double forward, double stdDev, double discount) {
strike_ = payoff.strike();
forward_ = forward;
stdDev_ = stdDev;
discount_ = discount;
variance_ = stdDev*stdDev;
if (!(forward>0.0))
throw new ApplicationException("positive forward value required: " + forward + " not allowed");
if (!(stdDev>=0.0))
throw new ApplicationException("non-negative standard deviation required: " + stdDev + " not allowed");
if (!(discount>0.0))
throw new ApplicationException("positive discount required: " + discount + " not allowed");
if (stdDev_>=Const.QL_EPSILON) {
if (strike_==0.0) {
n_d1_ = 0.0;
n_d2_ = 0.0;
cum_d1_ = 1.0;
cum_d2_= 1.0;
} else {
D1_ = Math.Log(forward/strike_)/stdDev_ + 0.5*stdDev_;
D2_ = D1_-stdDev_;
CumulativeNormalDistribution f = new CumulativeNormalDistribution();
cum_d1_ = f.value(D1_);
cum_d2_= f.value(D2_);
n_d1_ = f.derivative(D1_);
n_d2_ = f.derivative(D2_);
}
} else {
if (forward>strike_) {
cum_d1_ = 1.0;
cum_d2_= 1.0;
} else {
cum_d1_ = 0.0;
cum_d2_= 0.0;
}
n_d1_ = 0.0;
n_d2_ = 0.0;
}
X_ = strike_;
DXDstrike_ = 1.0;
// the following one will probably disappear as soon as
// super-share will be properly handled
DXDs_ = 0.0;
// this part is always executed.
// in case of plain-vanilla payoffs, it is also the only part
// which is executed.
switch (payoff.optionType()) {
case Option.Type.Call:
alpha_ = cum_d1_;// N(d1)
DalphaDd1_ = n_d1_;// n(d1)
beta_ = -cum_d2_;// -N(d2)
DbetaDd2_ = - n_d2_;// -n(d2)
break;
case Option.Type.Put:
alpha_ = -1.0+cum_d1_;// -N(-d1)
DalphaDd1_ = n_d1_;// n( d1)
beta_ = 1.0-cum_d2_;// N(-d2)
DbetaDd2_ = - n_d2_;// -n( d2)
break;
default:
throw new ArgumentException("invalid option type");
}
// now dispatch on type.
Calculator calc = new Calculator(this);
payoff.accept(calc);
}
void REPORT_FAILURE(string greekName, StrikedTypePayoff payoff, Exercise exercise, double s, double q, double r,
Date today, double v, double expected, double calculated, double error, double tolerance)
{
Assert.Fail(exercise + " "
+ payoff.optionType() + " option with "
+ payoff + " payoff:\n"
+ " spot value: " + s + "\n"
+ " strike: " + payoff.strike() + "\n"
+ " dividend yield: " + q + "\n"
+ " risk-free rate: " + r + "\n"
+ " reference date: " + today + "\n"
+ " maturity: " + exercise.lastDate() + "\n"
+ " volatility: " + v + "\n\n"
+ " expected " + greekName + ": " + expected + "\n"
+ " calculated " + greekName + ": " + calculated + "\n"
+ " error: " + error + "\n"
+ " tolerance: " + tolerance);
}
// critical commodity price
public static double criticalPrice(StrikedTypePayoff payoff, double riskFreeDiscount, double dividendDiscount,
double variance, double tolerance)
{
// Calculation of seed value, Si
double n = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / (variance);
double m = -2.0 * Math.Log(riskFreeDiscount) / (variance);
double bT = Math.Log(dividendDiscount / riskFreeDiscount);
double qu, Su, h, Si = 0;
switch (payoff.optionType())
{
case Option.Type.Call:
qu = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * m)) / 2.0;
Su = payoff.strike() / (1.0 - 1.0 / qu);
h = -(bT + 2.0 * Math.Sqrt(variance)) * payoff.strike() /
(Su - payoff.strike());
Si = payoff.strike() + (Su - payoff.strike()) *
(1.0 - Math.Exp(h));
break;
case Option.Type.Put:
qu = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * m)) / 2.0;
Su = payoff.strike() / (1.0 - 1.0 / qu);
h = (bT - 2.0 * Math.Sqrt(variance)) * payoff.strike() /
(payoff.strike() - Su);
Si = Su + (payoff.strike() - Su) * Math.Exp(h);
break;
default:
Utils.QL_FAIL("unknown option type");
break;
}
// Newton Raphson algorithm for finding critical price Si
double Q, LHS, RHS, bi;
double forwardSi = Si * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance) /
Math.Sqrt(variance);
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double K = (!Utils.close(riskFreeDiscount, 1.0, 1000))
? -2.0 * Math.Log(riskFreeDiscount)
/ (variance * (1.0 - riskFreeDiscount))
: 2.0 / variance;
double temp = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
switch (payoff.optionType())
{
case Option.Type.Call:
Q = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4 * K)) / 2;
LHS = Si - payoff.strike();
RHS = temp + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q) +
(1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS) / payoff.strike() > tolerance)
{
Si = (payoff.strike() + RHS - bi * Si) / (1 - bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
LHS = Si - payoff.strike();
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
RHS = temp2 + (1 - dividendDiscount * cumNormalDist.value(d1)) * Si / Q;
bi = dividendDiscount * cumNormalDist.value(d1) * (1 - 1 / Q)
+ (1 - dividendDiscount *
cumNormalDist.derivative(d1) / Math.Sqrt(variance))
/ Q;
}
break;
case Option.Type.Put:
Q = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4 * K)) / 2;
LHS = payoff.strike() - Si;
RHS = temp - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount *cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
while (Math.Abs(LHS - RHS) / payoff.strike() > tolerance)
{
Si = (payoff.strike() - RHS + bi * Si) / (1 + bi);
forwardSi = Si * dividendDiscount / riskFreeDiscount;
d1 = (Math.Log(forwardSi / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
LHS = payoff.strike() - Si;
double temp2 = Utils.blackFormula(payoff.optionType(), payoff.strike(),
forwardSi, Math.Sqrt(variance)) * riskFreeDiscount;
RHS = temp2 - (1 - dividendDiscount * cumNormalDist.value(-d1)) * Si / Q;
bi = -dividendDiscount *cumNormalDist.value(-d1) * (1 - 1 / Q)
- (1 + dividendDiscount * cumNormalDist.derivative(-d1)
/ Math.Sqrt(variance)) / Q;
}
break;
default:
Utils.QL_FAIL("unknown option type");
break;
}
return(Si);
}
public override void calculate()
{
Utils.QL_REQUIRE(arguments_.exercise.type() == Exercise.Type.American, () => "not an American Option");
AmericanExercise ex = arguments_.exercise as AmericanExercise;
Utils.QL_REQUIRE(ex != null, () => "non-American exercise given");
Utils.QL_REQUIRE(!ex.payoffAtExpiry(), () => "payoff at expiry not handled");
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
Utils.QL_REQUIRE(payoff != null, () => "non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
Utils.QL_REQUIRE(spot > 0.0, () => "negative or null underlying given");
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount >= 1.0 && payoff.optionType() == Option.Type.Call)
{
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
}
else
{
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
double tolerance = 1e-6;
double Sk = criticalPrice(payoff, riskFreeDiscount,
dividendDiscount, variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double d1 = (Math.Log(forwardSk / payoff.strike()) + 0.5 * variance)
/ Math.Sqrt(variance);
double n = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / variance;
double K = (!Utils.close(riskFreeDiscount, 1.0, 1000))
? -2.0 * Math.Log(riskFreeDiscount)
/ (variance * (1.0 - riskFreeDiscount))
: 2.0 / variance;
double Q, a;
switch (payoff.optionType())
{
case Option.Type.Call:
Q = (-(n - 1.0) + Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * K)) / 2.0;
a = (Sk / Q) * (1.0 - dividendDiscount * cumNormalDist.value(d1));
if (spot < Sk)
{
results_.value = black.value() +
a * Math.Pow((spot / Sk), Q);
}
else
{
results_.value = spot - payoff.strike();
}
break;
case Option.Type.Put:
Q = (-(n - 1.0) - Math.Sqrt(((n - 1.0) * (n - 1.0)) + 4.0 * K)) / 2.0;
a = -(Sk / Q) *
(1.0 - dividendDiscount * cumNormalDist.value(-d1));
if (spot > Sk)
{
results_.value = black.value() +
a * Math.Pow((spot / Sk), Q);
}
else
{
results_.value = payoff.strike() - spot;
}
break;
default:
Utils.QL_FAIL("unknown option type");
break;
}
} // end of "early exercise can be optimal"
}
public override void calculate()
{
Utils.QL_REQUIRE(arguments_.exercise.type() == Exercise.Type.American, () => "not an American Option");
AmericanExercise ex = arguments_.exercise as AmericanExercise;
Utils.QL_REQUIRE(ex != null, () => "non-American exercise given");
Utils.QL_REQUIRE(!ex.payoffAtExpiry(), () => "payoff at expiry not handled");
StrikedTypePayoff payoff = arguments_.payoff as StrikedTypePayoff;
Utils.QL_REQUIRE(payoff != null, () => "non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(ex.lastDate(), payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(ex.lastDate());
double riskFreeDiscount = process_.riskFreeRate().link.discount(ex.lastDate());
double spot = process_.stateVariable().link.value();
Utils.QL_REQUIRE(spot > 0.0, () => "negative or null underlying given");
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
if (dividendDiscount >= 1.0 && payoff.optionType() == Option.Type.Call)
{
// early exercise never optimal
results_.value = black.value();
results_.delta = black.delta(spot);
results_.deltaForward = black.deltaForward();
results_.elasticity = black.elasticity(spot);
results_.gamma = black.gamma(spot);
DayCounter rfdc = process_.riskFreeRate().link.dayCounter();
DayCounter divdc = process_.dividendYield().link.dayCounter();
DayCounter voldc = process_.blackVolatility().link.dayCounter();
double t = rfdc.yearFraction(process_.riskFreeRate().link.referenceDate(), arguments_.exercise.lastDate());
results_.rho = black.rho(t);
t = divdc.yearFraction(process_.dividendYield().link.referenceDate(), arguments_.exercise.lastDate());
results_.dividendRho = black.dividendRho(t);
t = voldc.yearFraction(process_.blackVolatility().link.referenceDate(), arguments_.exercise.lastDate());
results_.vega = black.vega(t);
results_.theta = black.theta(spot, t);
results_.thetaPerDay = black.thetaPerDay(spot, t);
results_.strikeSensitivity = black.strikeSensitivity();
results_.itmCashProbability = black.itmCashProbability();
}
else
{
// early exercise can be optimal
CumulativeNormalDistribution cumNormalDist = new CumulativeNormalDistribution();
NormalDistribution normalDist = new NormalDistribution();
double tolerance = 1e-6;
double Sk = BaroneAdesiWhaleyApproximationEngine.criticalPrice(payoff, riskFreeDiscount, dividendDiscount,
variance, tolerance);
double forwardSk = Sk * dividendDiscount / riskFreeDiscount;
double alpha = -2.0 * Math.Log(riskFreeDiscount) / (variance);
double beta = 2.0 * Math.Log(dividendDiscount / riskFreeDiscount) / (variance);
double h = 1 - riskFreeDiscount;
double phi = 0;
switch (payoff.optionType())
{
case Option.Type.Call:
phi = 1;
break;
case Option.Type.Put:
phi = -1;
break;
default:
Utils.QL_FAIL("invalid option type");
break;
}
//it can throw: to be fixed
// FLOATING_POINT_EXCEPTION
double temp_root = Math.Sqrt((beta - 1) * (beta - 1) + (4 * alpha) / h);
double lambda = (-(beta - 1) + phi * temp_root) / 2;
double lambda_prime = -phi * alpha / (h * h * temp_root);
double black_Sk = Utils.blackFormula(payoff.optionType(), payoff.strike(), forwardSk, Math.Sqrt(variance)) *
riskFreeDiscount;
double hA = phi * (Sk - payoff.strike()) - black_Sk;
double d1_Sk = (Math.Log(forwardSk / payoff.strike()) + 0.5 * variance) / Math.Sqrt(variance);
double d2_Sk = d1_Sk - Math.Sqrt(variance);
double part1 = forwardSk * normalDist.value(d1_Sk) / (alpha * Math.Sqrt(variance));
double part2 = -phi *forwardSk *cumNormalDist.value(phi *d1_Sk) * Math.Log(dividendDiscount) /
Math.Log(riskFreeDiscount);
double part3 = +phi *payoff.strike() * cumNormalDist.value(phi * d2_Sk);
double V_E_h = part1 + part2 + part3;
double b = (1 - h) * alpha * lambda_prime / (2 * (2 * lambda + beta - 1));
double c = -((1 - h) * alpha / (2 * lambda + beta - 1)) *
(V_E_h / (hA) + 1 / h + lambda_prime / (2 * lambda + beta - 1));
double temp_spot_ratio = Math.Log(spot / Sk);
double chi = temp_spot_ratio * (b * temp_spot_ratio + c);
if (phi * (Sk - spot) > 0)
{
results_.value = black.value() + hA * Math.Pow((spot / Sk), lambda) / (1 - chi);
}
else
{
results_.value = phi * (spot - payoff.strike());
}
double temp_chi_prime = (2 * b / spot) * Math.Log(spot / Sk);
double chi_prime = temp_chi_prime + c / spot;
double chi_double_prime = 2 * b / (spot * spot) - temp_chi_prime / spot - c / (spot * spot);
results_.delta = phi * dividendDiscount * cumNormalDist.value(phi * d1_Sk) +
(lambda / (spot * (1 - chi)) + chi_prime / ((1 - chi) * (1 - chi))) *
(phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot / Sk), lambda);
results_.gamma = phi * dividendDiscount * normalDist.value(phi * d1_Sk) / (spot * Math.Sqrt(variance)) +
(2 * lambda * chi_prime / (spot * (1 - chi) * (1 - chi)) +
2 * chi_prime * chi_prime / ((1 - chi) * (1 - chi) * (1 - chi)) +
chi_double_prime / ((1 - chi) * (1 - chi)) +
lambda * (1 - lambda) / (spot * spot * (1 - chi))) *
(phi * (Sk - payoff.strike()) - black_Sk) * Math.Pow((spot / Sk), lambda);
} // end of "early exercise can be optimal"
}